How to properly scale image in uv editor
You don't need the eye in the whole eyeball, so select only the faces that you want the image on.
In the case of a UV sphere it's easier to do this in top orhto view.
Then do project from view (bounds).

Use that UV map to map thimage on a material for the iris,
then assign a different material for the rest of the eye.

Note that the second material would not be necessary if the original image had an alpha channel filling the image around the iris instead of just white, then you could use the alpha to have better control on the placement of the texture.